direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C15, C4.C30, C60.7C2, C20.3C6, C12.3C10, C30.24C22, C10.7(C2×C6), C2.2(C2×C30), C6.7(C2×C10), SmallGroup(120,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C15
G = < a,b,c | a15=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C15
►Regular action on 120 points
Generators in S120
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15)(16 17 18 19 20 21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75)(76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100 101 102 103 104 105)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 104 37 75)(2 105 38 61)(3 91 39 62)(4 92 40 63)(5 93 41 64)(6 94 42 65)(7 95 43 66)(8 96 44 67)(9 97 45 68)(10 98 31 69)(11 99 32 70)(12 100 33 71)(13 101 34 72)(14 102 35 73)(15 103 36 74)(16 52 88 115)(17 53 89 116)(18 54 90 117)(19 55 76 118)(20 56 77 119)(21 57 78 120)(22 58 79 106)(23 59 80 107)(24 60 81 108)(25 46 82 109)(26 47 83 110)(27 48 84 111)(28 49 85 112)(29 50 86 113)(30 51 87 114)
(1 116 37 53)(2 117 38 54)(3 118 39 55)(4 119 40 56)(5 120 41 57)(6 106 42 58)(7 107 43 59)(8 108 44 60)(9 109 45 46)(10 110 31 47)(11 111 32 48)(12 112 33 49)(13 113 34 50)(14 114 35 51)(15 115 36 52)(16 103 88 74)(17 104 89 75)(18 105 90 61)(19 91 76 62)(20 92 77 63)(21 93 78 64)(22 94 79 65)(23 95 80 66)(24 96 81 67)(25 97 82 68)(26 98 83 69)(27 99 84 70)(28 100 85 71)(29 101 86 72)(30 102 87 73)
G:=sub<Sym(120)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,104,37,75)(2,105,38,61)(3,91,39,62)(4,92,40,63)(5,93,41,64)(6,94,42,65)(7,95,43,66)(8,96,44,67)(9,97,45,68)(10,98,31,69)(11,99,32,70)(12,100,33,71)(13,101,34,72)(14,102,35,73)(15,103,36,74)(16,52,88,115)(17,53,89,116)(18,54,90,117)(19,55,76,118)(20,56,77,119)(21,57,78,120)(22,58,79,106)(23,59,80,107)(24,60,81,108)(25,46,82,109)(26,47,83,110)(27,48,84,111)(28,49,85,112)(29,50,86,113)(30,51,87,114), (1,116,37,53)(2,117,38,54)(3,118,39,55)(4,119,40,56)(5,120,41,57)(6,106,42,58)(7,107,43,59)(8,108,44,60)(9,109,45,46)(10,110,31,47)(11,111,32,48)(12,112,33,49)(13,113,34,50)(14,114,35,51)(15,115,36,52)(16,103,88,74)(17,104,89,75)(18,105,90,61)(19,91,76,62)(20,92,77,63)(21,93,78,64)(22,94,79,65)(23,95,80,66)(24,96,81,67)(25,97,82,68)(26,98,83,69)(27,99,84,70)(28,100,85,71)(29,101,86,72)(30,102,87,73)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,104,37,75)(2,105,38,61)(3,91,39,62)(4,92,40,63)(5,93,41,64)(6,94,42,65)(7,95,43,66)(8,96,44,67)(9,97,45,68)(10,98,31,69)(11,99,32,70)(12,100,33,71)(13,101,34,72)(14,102,35,73)(15,103,36,74)(16,52,88,115)(17,53,89,116)(18,54,90,117)(19,55,76,118)(20,56,77,119)(21,57,78,120)(22,58,79,106)(23,59,80,107)(24,60,81,108)(25,46,82,109)(26,47,83,110)(27,48,84,111)(28,49,85,112)(29,50,86,113)(30,51,87,114), (1,116,37,53)(2,117,38,54)(3,118,39,55)(4,119,40,56)(5,120,41,57)(6,106,42,58)(7,107,43,59)(8,108,44,60)(9,109,45,46)(10,110,31,47)(11,111,32,48)(12,112,33,49)(13,113,34,50)(14,114,35,51)(15,115,36,52)(16,103,88,74)(17,104,89,75)(18,105,90,61)(19,91,76,62)(20,92,77,63)(21,93,78,64)(22,94,79,65)(23,95,80,66)(24,96,81,67)(25,97,82,68)(26,98,83,69)(27,99,84,70)(28,100,85,71)(29,101,86,72)(30,102,87,73) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75),(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,104,37,75),(2,105,38,61),(3,91,39,62),(4,92,40,63),(5,93,41,64),(6,94,42,65),(7,95,43,66),(8,96,44,67),(9,97,45,68),(10,98,31,69),(11,99,32,70),(12,100,33,71),(13,101,34,72),(14,102,35,73),(15,103,36,74),(16,52,88,115),(17,53,89,116),(18,54,90,117),(19,55,76,118),(20,56,77,119),(21,57,78,120),(22,58,79,106),(23,59,80,107),(24,60,81,108),(25,46,82,109),(26,47,83,110),(27,48,84,111),(28,49,85,112),(29,50,86,113),(30,51,87,114)], [(1,116,37,53),(2,117,38,54),(3,118,39,55),(4,119,40,56),(5,120,41,57),(6,106,42,58),(7,107,43,59),(8,108,44,60),(9,109,45,46),(10,110,31,47),(11,111,32,48),(12,112,33,49),(13,113,34,50),(14,114,35,51),(15,115,36,52),(16,103,88,74),(17,104,89,75),(18,105,90,61),(19,91,76,62),(20,92,77,63),(21,93,78,64),(22,94,79,65),(23,95,80,66),(24,96,81,67),(25,97,82,68),(26,98,83,69),(27,99,84,70),(28,100,85,71),(29,101,86,72),(30,102,87,73)]])
Q8×C15 is a maximal subgroup of
Q8⋊2D15 C15⋊7Q16 Q8⋊3D15
75 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 10A | 10B | 10C | 10D | 12A | ··· | 12F | 15A | ··· | 15H | 20A | ··· | 20L | 30A | ··· | 30H | 60A | ··· | 60X |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | - | |||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 | Q8 | C3×Q8 | C5×Q8 | Q8×C15 |
| kernel | Q8×C15 | C60 | C5×Q8 | C3×Q8 | C20 | C12 | Q8 | C4 | C15 | C5 | C3 | C1 |
| # reps | 1 | 3 | 2 | 4 | 6 | 12 | 8 | 24 | 1 | 2 | 4 | 8 |
Matrix representation of Q8×C15 ►in GL2(𝔽31) generated by
| 20 | 0 |
| 0 | 20 |
| 27 | 17 |
| 30 | 4 |
| 0 | 18 |
| 12 | 0 |
G:=sub<GL(2,GF(31))| [20,0,0,20],[27,30,17,4],[0,12,18,0] >;
Q8×C15 in GAP, Magma, Sage, TeX
Q_8\times C_{15} % in TeX
G:=Group("Q8xC15"); // GroupNames label
G:=SmallGroup(120,33);
// by ID
G=gap.SmallGroup(120,33);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-2,300,621,306]);
// Polycyclic
G:=Group<a,b,c|a^15=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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